In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.

Is there any interpretation of these operators in terms of Grothendieck (or other) style geometric ideas such as thickenings, formal schemes, lambda rings, or crystalline cohomology?

A related and possibly equivalent question is whether there is any precise sense in which the "extra" material in divided power algebras (compared to polynomial or power-series rings) is dual to the "extra" differential operators in positive characteristic(s).

  • $\begingroup$ Could you make the question a little bit more precise, I don't really know what you are looking for. Differential operators are defined in terms of thickenings: The sheaf of operators of order <= n on X is dual to the structure sheaf of the n-th infinitesimal neighborhood of the diagonal of X in X x X. Similarly, the sheaf of all diff ops is dual to the structure sheaf of the formal nbhd. of the diagonal. For crystalline theory you add divided power structures, which make the extra operators go away. Ring of PD-DiffOps=ring gen'd by derivations and O_X if I remember correctly. $\endgroup$ – Lars Jun 24 '10 at 6:08
  • $\begingroup$ But why is the PD structure the correct modification (other than "it works" in making the perceived "extras" go away)? To put it differently, is there some canonical geometric meaning to the subring of differential operators generated by derivations? The same coefficients appear in the Leibniz rule for degree $n$ derivations and in the PD structure, so possibly it is just forced algebraically, but is that actually the case or it only makes the PD enlargement one of many possible options? $\endgroup$ – CFZ Jun 24 '10 at 7:44
  • $\begingroup$ Using the composition formula, one sees that in order to write a given operator of order n as product of derivations you have to be able to divide by n!. That's what's not possible in pos. char., but divided power structures restore this ability. For a geometric meaning: The modules over the subring gen'd by deriviations are precisely the integrable connections of p-curvature 0. Is that what you're looking for? I can recommend the first chapter of Berthelot-Ogus' "Notes on Crystalline Cohomology". $\endgroup$ – Lars Jun 24 '10 at 8:21

In a nutshell, observe that $\partial_x^p(x^p)=0$ in characteristic $p$. This means that you can divide either $\partial_x^p$ or $x^p$ by $p!$ and get a sensible object, i.e., the second object will interact with the divided object and you get $=1$ in the formula. Dividing $\partial_x^p$ you get your missing differential operators. Dividing $x^p$ you lose it but discover "crystalline world", "sickening", etc.

To put it bluntly, No, in crystalline world, differential operators on a smooth scheme are generated in degree 1, but yes, your missing operators will live on a formal scheme...

What was your question exactly, Doc?


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